Discrete Jackson q-Linearization of the Kaprekar Kernel: $3$-Adic Factorial Isomorphism, Causal Continuum Limits, and Spectral Models for Quadratic Irrationals

Motivated by Borel's open problem on the normal distribution of digits of quadratic irrationals on the real line R, we develop an operator-theoretic framework connecting non-Archimedean Kaprekar dynamics with discrete quantum q-calculus ($q = 1/b = 1/10$). To model the suppression of non-local Archimedean digit carries along the periodic continued fraction carrier of √2 - 1 = [0; (2)], we construct a q-deformed dynamic kernel regulated by an infinite-state S₃-valued digit transducer. Our primary algebraic result establishes an exact $3$-adic valuation isomorphism for Jackson q-factorials, ν₃([n]q!) = ν₃(n!), proving that discrete q-quantization introduces no small denominator obstructions or hidden Galois phase shifts. Using this invariance, we prove the existence and uniqueness of a regular Hensel root Sq^*(Δ) ∈ 1 + 9Z₃ via a convergent quantum Bernoulli--Taylor series. In the continuum limit q → 1⁻, we establish the weak Prokhorov convergence of the resulting causal model measures to the Lebesgue measure under uniform Weyl remainder assumptions, verify boundary non-atomicity via Liouville's Diophantine inequality, and demonstrate structural compatibility with the positive spectral gap of the Gauss--Kuzmin--Wirsing transfer operator. Finally, we formulate open hypotheses regarding full Borel normality on R and higher-degree radical correspondences for sparse generating functions F_α(t) (α ≥ 3).